UQF-5C — Interacting / strong-coupling graviton (QG Q1 strong half): full dossier

Binding status (matches the live popup, never upgraded): OPEN (global wall) — honestly disclosed, not promotable. Direction: held. STATUS-UPGRADES:0. Frozen geometry dcc66f1b2685 / a5b1e6f9d951 READ-ONLY.

This dossier expands the 30-second gate brief into a working-physicist-depth treatment of the strong-coupling graviton gate. It is written to be checked and built on: every number traces to a corpus artifact named in §5, and the closure section (§6) is a concrete specialist work plan, not a wish-list. The cardinal discipline is carried verbatim throughout: a₆ is the missing object for UV completion, never is it; relabeling "a₆ computed" as "5C closed" relocates the wall and is refused.

1. Executive summary + honest status

Headline. The strong-coupling graviton is openly unfinished here — and it is unfinished for every theory of quantum gravity, so this is the field's wall, not a defect of ours. The frozen 13D geometry hands you the graviton operator cleanly; it supplies no new lever to cross the strong-coupling wall, and we say so flat-out.

The honest grade. UQF-5C is OPEN (global wall). This is not a hedge and it is not a failure: a full UV-complete, strong-coupling quantum theory of gravity — gravitons scattering hard above the cutoff — is one of the great open problems in physics, unsolved by strings, by asymptotic safety, by loops, and by us. The honest move is to state the wall precisely, locate exactly where our construction stops, name the missing object, and lay out what a specialist would actually do next. Where we have banked genuine, non-promoting results — the geometry-supplied operator as structure, the proved curvature ratio |Riem|²/R² = 23/75, the scope firewall, the T-DEEP no-finite-grain-shortcut result, and a decision-grade refutation of a wrong magnitude — we present them as exactly that: real progress that does not certify the quantum theory.

What this dossier establishes and does not. It establishes (a) that the frozen geometry supplies the graviton operator (de-Donder gauge-fixed metric fluctuation + Faddeev–Popov ghost sector; 5A as structure, given-E); (b) that the geometric curvature input |Riem|²/R² = 23/75 is pinned exactly by independent routes (a banked input win on the leg below, not a gate advance); (c) that the scope firewall — even a finite, positive heat-kernel coefficient a₆ certifies one consistency coefficient, never a UV completion — is correct and terminal as a scope wall; and (d) that a wrong dimensionful magnitude (the corpus bulk −2.818×10⁹⁴ GeV⁶) was refuted at decision grade for the graviton sector rather than shipped. It does not establish — and explicitly does not claim — a UV completion, a strong-coupling closure, a derivation of the interacting graviton S-matrix, or that this geometry is the unique one that could supply a consistent spin-2 sector. The interacting / strong-coupling leg (5C) is OPEN and stays OPEN; the linearized legs one floor below (5A/5B) are CERTIFICATE-CONDITIONAL (conditional on UQF-9), and we never let a 5B fact masquerade as a 5C closure.

The one-sentence endpoint. The strong-coupling graviton gate reduces to a single, precisely-named global wall — a constructive, target-blind, non-perturbative UV completion of the interacting graviton on the frozen geometry (UQF-9) — for which the geometry supplies the operator (the de-Donder L_grav + FP-ghost sector, terminal as structure, given-E) but no constructive lever; everything that would certify the quantum theory (the finite a₆ vector and its positivity test on the leg below, the asymptotic-safety fixed point on this leg) stays OPEN, so the gate holds at the honest ceiling "serious candidate / the geometry supplies the operator, NOT a validated quantum theory of gravity," with the only decision-grade move available being a real, reproducibility-gated a₆-positivity computation one leg below — never a relabeling of "a₆ computed" as "5C closed."

2. The community gap

2.1 The precise open problem

Quantum gravity has two qualitatively different halves, and this gate is the hard half. Write the metric as a background plus a fluctuation, g = ḡ + h. The linearized (weak-field) theory — the propagating spin-2 graviton h, its two massless helicities, the Newtonian and linearized-Einstein limits, and the one-loop heat-kernel consistency of that quadratic action — is leg 5A/5B and lives one floor below this dossier. The interacting, strong-coupling theory — gravitons scattering hard at and above the Planck/cutoff scale, where the cubic and higher vertices of the Einstein–Hilbert action dominate, where the perturbative series in G_N E² ceases to converge, and where the theory must either exhibit a genuine high-energy completion or replace "all the way to infinite energy" with something defensible — is 5C. That is the object this gate is about.

The community statement is blunt: no program has a finished, agreed strong-coupling/UV-complete quantum theory of gravity. Perturbative quantized general relativity is non-renormalizable (the two-loop counterterm of 't Hooft–Veltman / Goroff–Sagnotti is the canonical demonstration that new divergences appear order by order with no closing pattern). Every serious approach is a bet on a mechanism to cross that wall, and none has been carried to a community-accepted completion.

2.2 State of the art / the best the field has

• String theory replaces the point graviton with an extended object; it is UV-finite order by order in many amplitudes, but it does not exhibit a single selected 4D vacuum (the landscape problem), and "the" non-perturbative definition (M-theory) is itself incomplete.
• Asymptotic safety bets on a non-Gaussian (interacting) UV fixed point of the gravitational renormalization group. The functional-renormalization-group (FRG) evidence for such a fixed point exists only at a declared truncation of the effective action; a truncation-independent existence proof is not in hand, and the trend of the evidence is contested. This is precisely the object our UQF-9 gate names.
• Loop quantum gravity / spin foams quantize geometry non-perturbatively but do not yet deliver a controlled weak-field graviton S-matrix matching perturbative GR in a settled way.
• Effective field theory treats GR as a perfectly predictive low-energy EFT — which is correct and uncontroversial below the cutoff, and is exactly the regime where leg 5A/5B lives — but the EFT explicitly declines the strong-coupling question; it is the statement of the wall, not a crossing of it.

So the honest "best bound" is: the linearized theory is under control as an EFT; the strong-coupling completion is open everywhere. This gate inherits that landscape exactly.

2.3 Where prior attempts (including ours) fall short

Our geometry's specific contribution and its specific limit:

• What the geometry does. On the frozen background M₄ × K₆ × S² × S¹_Y/ℤ₂ (K₆ = SU(3)/T², M,N = 1…13), the metric fluctuation h_MN gauge-fixed in de-Donder (harmonic) gauge yields a standard Lichnerowicz-type Laplace operator L_grav = −(∇² + E). Its 4D massless spin-2 mode is the graviton, with the Kaluza–Klein tower above. The geometry supplies the operator whose properties decide consistency.
• What it does not do. Supplying the operator is the setup for a consistency question, not a crossing of the strong-coupling wall. The geometry provides no new non-perturbative lever, no controlled fixed point beyond a truncation, and no UV-complete S-matrix above the cutoff. It therefore cannot, by itself, certify that the quantum theory is finite or unitary in the strong-coupling regime.

This is the gate's load-bearing honesty: a Kaluza–Klein GUT candidate that carries a graviton is a real and non-trivial structural fact, but it is not a UV completion, and the corpus refuses to let the two be conflated.

3. The construction — rigorous math

This section shows the actual machinery: how the geometry supplies the operator, what the heat-kernel consistency ladder is, exactly which object (a₆) lives on the leg below, what numbers are genuinely in hand and how they were computed, and — critically for this gate — exactly where the construction stops short of 5C.

3.1 The graviton operator from the frozen geometry (5A, as structure, given-E)

The spine of a heat-kernel consistency computation is fixed and standard:

OPERATOR  →  UNIVERSAL FORMULA  →  SPECIFIC TRACE  →  INTERPRETATION

Concretely, on the frozen background:

h_MN on M₄×K₆×S²×S¹_Y   --de-Donder gauge-fix-->   L_grav = −(∇² + E)   (Lichnerowicz-type)
       + FP-ghost sector  --GHOST-CORRECTED-->      physically meaningful operator
       --Gilkey/Avramidi a₆ template (d=13, cubic-curvature basis)-->   tr[a₆]    [NOT COMPUTED]
       --positivity functional P(a₆) ≥ 0 ?-->        weak-field UV consistency (5B)  [P UNSELECTED]

The de-Donder graviton operator and the Faddeev–Popov ghost sector are standard textbook structures (the form is generic to any spin-2 quantization; the d=13 specialization is geometry-indexed). The 4D massless spin-2 mode is the graviton; the heavy KK tower rides above. This is 5A, terminal as structure, given-E — it reproduces linearized GR given the observed gravity limit as input, not as a from-nothing derivation. (given-E ≠ derivation of E.) Source: GRAVITON_COMPLETION_HANDOFF/01_DOSSIER.md §1; GATE_BRIEF_GRAVITON.md.

The physically meaningful object is ghost-corrected. A graviton-only result is gauge-dependent and not even well-posed as a consistency claim. The physical coefficient is the ghost-corrected trace; in the de-Donder gauge the fiber trace is

graviton on Sym²T (dim 91) − 2 × FP vector ghost (dim 13) → physical combination = grav − 2·ghost = ×65.

The multiplicity 91 = dim Sym²(ℝ¹³) = 13·14/2 and the ghost dimension 13 are fixed by the de-Donder gauge structure; the −2 is the Faddeev–Popov sign+count. Source: computational_runs_2026-06-23/a6_computation/ GAP01_A6_RESULT_COMPLETE.md §1 ("Fiber trace … grav − 2·ghost = ×65"). The sign and multiplicity are forced by BRST nilpotency (well-posed), but they are not yet verified against an actually-computed graviton trace — that is residual R5 one floor below.

3.2 The heat-kernel consistency ladder and the named missing object a₆

For a Laplace-type operator L = −(∇² + E) on a d-dimensional manifold, the heat trace has the short-time expansion tr e^{−tL} ~ (4πt)^{−d/2} Σ_k a_{2k}(L) t^k, where the a_{2k} are the Seeley–DeWitt / heat-kernel coefficients — local curvature invariants integrated over the manifold. The universal functional for a₆ is Gilkey (1995), Theorem 4.8.16 = Vassilevich, Phys. Rept. 388 (2003) 279, eq. (4.29) (the closed-manifold a6nobou block), transcribed verbatim in the corpus from the arXiv TeX e-print. Source: GAP01_A6_RESULT_COMPLETE.md §1; A6_DERIVATIVE_SECTOR.md §0.

The relevance to gate consistency is the weak-field UV-consistency test (5B): assemble the d=13 graviton+ghost trace tr[a₆] over the cubic-curvature basis with K₆×S²×S¹_Y holonomy, and check it against a positivity functional P(a₆) ≥ 0 required for physical-Hilbert-space closure. tr[a₆] is not honestly computed (R1), and P is itself unselected (three competing readings, R4), so P(a₆) ≥ 0 is not yet a well-defined predicate. These two facts live on the leg below (5B), not in 5C — and this dossier is scrupulous about that boundary.

The scope firewall — why a₆ is not the wall-crosser (R7/S6). Even a flawless, finite, positive a₆ certifies one coefficient in an unbounded operator ladder a₆ < a₈ < a₁₀ < …. One coefficient cannot settle strong-coupling behavior. The corpus names a₆ as the missing object for UV completion, never as the completion. This is the gate's signature mis-close, and it is forbidden: relabeling "a₆ computed" as "5C closed" relocates the wall. Source: GAP01_A6_RESULT_COMPLETE.md §5 (MO-12: "a₆ is NECESSARY, NOT SUFFICIENT … one coefficient in the unbounded a8/a10/… tower"); 01_DOSSIER.md §3.1 R7.

3.3 The geometric curvature input |Riem|²/R² = 23/75 (banked, on the leg below)

A genuine, target-blind input win exists for the heat-kernel machinery, and the honesty of it is instructive. The corpus K₆ = SU(3)/T² curvature builder carried a sign error on two of the naturally-reductive quarter-bracket terms in the Riemann tensor. The buggy tensor violated the first Bianchi identity and produced

|Riem|²/Scal² = 31/147 = 0.21088 (buggy, Bianchi-violating)

instead of the correct, Bianchi-exact value

|Riem|²/Scal² = 23/75 = 0.30667 (corrected).

This is the ~31% curvature deficit (R2) referenced throughout the gate's documents: 0.2109 vs the spectrum-confirmed 0.3067 — a real, quantified, internal error, located precisely, not a hand-wave. Source: certificates/CERT_CURVATURE_DELTA_2026-06-25.md.

The root cause: the engine's k6_riem_gilkey::Rop() used +¼[[Y,Z]_𝔪,X]_𝔪 − ¼[[X,Z]_𝔪,Y]_𝔪, whereas the correct naturally-reductive form (Besse 7.38 / Kobayashi–Nomizu Vol. II) is the opposite sign on those two terms. The engine self-check missed it because it tested only Ricci isotropy and Scal/Ric = 6 — both insensitive to the sign error, since the Ricci contraction is unchanged. The decisive target-blind criterion is the first Bianchi identity itself, a theorem every Riemann tensor must satisfy:

Source: CERT_CURVATURE_DELTA_2026-06-25.md ("Numbers" table). The value 23/75 was independently reproduced by four target-blind methods (Nomizu Levi-Civita; covariant 2nd-Bianchi build; the corpus's own finite-difference reference k6_fd_nablaR.py; sectional-curvature structure sums) plus a from-scratch rebuild with a different inner product — all agreeing exactly, none touching any a₆ value. The 2026-06-25 consolidated view records this as |Riem|²/R² = 23/75 PROVED by independent routes (the live UQF-5C brief credits two independent exact routes: an SU(3)/T² structure-constant computation a6_07_k6_indep.py and the spectrum-demanded value from the exact heat-trace a4). Source: GATE_REGRADE_BESTCASE_AND_HOLES_2026-06-29.md UQF-5C "Proven (banked)" + reviewer note.

Honest caveat (carried, not buried). Correcting a curvature input is necessary, not sufficient. It does not close the gate. And the live status of R2 is contested between same-date frozen docs: the canonical baseline (GAP01.md) holds the ~31% error as a LIVE, un-discharged computation-debt that "poisons every downstream magnitude," not a resolved one. We adopt the conservative reading: the corrected value 23/75 is proved; the engine fix is owed; no downstream magnitude is yet trustworthy.

3.4 The dimensionless color ratio 124/315 (DERIVED-PENDING, on the leg below)

The genuinely scale-free heart of the a₆ color factor is the dimensionless ratio 124/315, accompanied by four exact-rational round-sphere cross-checks of the algebraic functional:

S² = 4/315, S⁴ = 74/63, S⁶ = 1139/63, S⁶(conformal) = 5/63 (exact).

Source: certificates/CERT_SCALEFREE_SECTOR_2026-06-25.md ("Claim"); GAP01_A6_RESULT_COMPLETE.md §2a. These sphere checks reproduce the actual Sⁿ eigenvalue heat trace on the spectral side, independently of the d=13 magnitude — they are not back-fitted (the verifier confirms no 2.8/94/target tokens steered them).

The honest downgrade (load-bearing for this gate). The 2026-06-25 consolidated audit downgraded 124/315 to DERIVED-PENDING-INDEPENDENT-TARGET-BLIND-REPRODUCTION, because the same R2-carrying engine produced it. Until 124/315 is reproduced by an independent computation that does not route through the (sign-bugged) engine, it is not to be sold as a settled, clean derived win. Source: GATE_REGRADE…2026-06-29.md UQF-5C bright-line flag; UQF-5A/5B holes ("124/315 … downgraded … same R2-carrying engine produced it"). A further honest tension is recorded in the leg-below handoff: GAP01.md notes 124/315 is metric-SELECTED (it holds at the spectrum-demanded Scal_K₆ = 7.5 normalization; the corpus's own declared curvature gives a different number), so the ratio carries that caveat and must not be advertised as the clean target-blind invariant. We carry it as exactly what it is: a strong, dual-checked, but not-yet-independently-reproduced scale-free result, one leg below 5C.

3.5 The derivative-of-curvature sector and the bulk magnitude (and why the magnitude is refuted)

For completeness of the heat-kernel object, the a₆ density splits into an algebraic (cubic-curvature) sector and a covariant-derivative-of-curvature (∇R) sector. On K₆ (homogeneous Einstein, non-symmetric), the scalar is constant and the Ricci is parallel, so (∇R)² = 0 and (∇Ric)² = 0; only (∇Riem)² survives, with the Gilkey/Vassilevich coefficient +9 × 1/7!. The measured invariants:

|∇Riem|² = 0.25 (unit normalization), ρ = |∇Riem|²/|Riem|³ = 0.094215, 2nd-Bianchi residual ~1.75×10⁻⁸ (FD precision).

Source: GAP01_A6_RESULT_COMPLETE.md §2b; A6_DERIVATIVE_SECTOR.md §3. The derivative machinery is cross-checked on the Berger (squashed) 3-sphere S³ = SU(2), g = diag(1,1,a²): round at a=1 gives ∇Riem = 0 exactly (a built-in null gate), and a ≠ 1 gives nonzero (∇Riem)² with exact 1st- and 2nd-Bianchi. (Honest disclosure carried verbatim from the corpus: a prior quantitative closed-form claim (∇Riem)² = 256·a²(a²−1)² was retracted when independent recomputation failed to reproduce it; the Berger check therefore validates the channel structurally, not against a published magnitude. Source: A6_DERIVATIVE_SECTOR.md §4, "RETRACTED".)

The engine emitted a complete bulk number, tr[a₆]_bulk = −2.817995812×10⁹⁴ GeV⁶. This dimensionful magnitude is REFUTED at decision grade for the graviton sector, for two independent reasons, and we refuse to ship it:

1. It is contaminated by R2. It was produced with the buggy 31/147 curvature; the corrected curvature shifts it by +6.305% to ≈ −2.9957×10⁹⁴ GeV⁶ (sign preserved). The originally-reported number is therefore wrong as a scale-free geometric prediction — its K₆ curvature normalization ran ~31% low. Source: CERT_CURVATURE_DELTA_2026-06-25.md; GATE_REGRADE…2026-06-29.md UQF-5C edge.
2. The dimensionful magnitude is scheme/scale-anchored, and arguably ill-posed at odd D=13 (R3). No canonical finite dimensionful a₆ exists at odd n: a₆ sits at the half-integer pole s = 7/2 (a power divergence, scheme-dependent, zero in dim-reg), and ζ_L(0) is holomorphic for odd n (no log / no anomaly slot). The banked GeV⁶ number is a local density × the dimensionless prefactor (4π)^{−13/2} = 7.1637×10⁻⁸, which is neither the canonical integrated A₆ (units GeV⁻⁷) nor an object that legitimately pairs with that prefactor. It rides one named, geometry-unfixed heat-kernel SCHEME-object — the same shared decision family as c_loop / SG-7-δ. Source: CERT_SCALEFREE_SECTOR_2026-06-25.md ("What is explicitly NOT derived").

So the only clean, scale-free content is the ratio (124/315, pending independent reproduction) and the geometric ratio (23/75, proved); the magnitude is not derived. This is the sharp split the gate insists on.

3.6 Where the construction stops short of 5C — the wall, stated precisely

Everything above is 5A/5B material — the operator (structure, given-E), the heat-kernel object on the leg below, and the input ratios. None of it crosses into 5C. The 5C wall is:

A constructive, target-blind, non-perturbative UV completion / strong-coupling closure of the interacting graviton on the frozen geometry. The geometry supplies the operator but exhibits no route across the wall.

This is the same object as UQF-9 (a truncation-independent non-Gaussian asymptotic-safety fixed point, or a defensible non-continuum replacement). Two downstream facts inherit from it:

• Graviton-sector unitarity above the cutoff (UQF-14 / R9) is inherited below the cutoff (standard QFT) but blocked by UQF-9 above it — there is no proof unitarity survives where the graviton and KK tower go strongly coupled until the UV completion exists.
• The a₆-positivity test (5B) is the nearest decision-grade move, but it lives one floor below and, even if it passes, lifts 5A/5B only to certificate-grade conditional on UQF-9 — it never reaches 5C (R7 firewall).

The T-DEEP no-finite-grain-shortcut result (banked). A natural hope is that the program's granularity / cost-floor axiom (which dissolves a class of continuum UV divergences on the leg toward UQF-9) might also supply a finite-grain shortcut to 5C. It does not. T-DEEP established that R₀ — the program's fundamental length — is a pure color/gauge object (R₀/l_Planck ~ 194), so every finite-grain dissolution of 5C relocates onto UQF-9 via the isolated sub-target P0 (does gravity own an intrinsic shortest length, or does it inherit the color R₀?). In short: 5C has no finite-grain shortcut; the wall is genuine. Source: GATE_REGRADE…2026-06-29.md UQF-5C "Proven (banked)" ("T-DEEP established R0 is a pure color/gauge object (R0/l_Planck ~ 194)").

4. The insights we used

These are the moves that made the honest progress believable and that a reader can reuse. They are shared in full; the firewall (no device engineering) is observed throughout.

4.1 Supply-the-operator, don't claim-the-theory

The single most clarifying move is the explicit separation of roles. The geometry's job is to supply the operator whose properties decide consistency — L_grav + the FP-ghost sector — and that is a real, geometry-indexed deliverable. Consistency is then a property of that operator, evaluated by standard heat-kernel machinery. Keeping these two roles distinct is what prevents the gate's signature mis-close: "we carry a graviton" is true and non-trivial; "we have a quantum theory of gravity" is false and forbidden. The whole 5A/5B/5C split is this discipline made structural.

4.2 The scope firewall: necessary ≠ sufficient, made into a theorem-shaped wall

The insight that a₆ is necessary-not-sufficient is not a hedge — it is a structural fact about the unbounded Seeley–DeWitt ladder a₆ < a₈ < a₁₀ < …. One coefficient, however clean, cannot certify an unbounded tower. This is the inverse-unicorn: the claim "one coefficient settles strong coupling" is itself the forbidden over-claim, so the firewall (R7) is the correct terminal disposition for the scope question, not an open weakness. It is what licenses us to compute a₆ honestly on the leg below without the result leaking upward into a false 5C closure.

4.3 First-Bianchi as the target-blind falsification test (how R2 was caught)

The curvature-deficit story is a model of target-blind discipline. The wrong tensor was caught not by comparison to any desired a₆ value, but by the first Bianchi identity — a theorem every Riemann tensor must satisfy, writable with zero reference to any target. The buggy tensor fails it (1/6); the corrected one passes (2.5×10⁻¹⁶). Four independent methods plus a different-inner-product rebuild all land on 23/75. This is the κ³/π discipline in action: the correction criterion is a theorem, not a match-to-known-answer, so it cannot be target-fitted. (And the discipline is symmetric: the resulting dimensionful magnitude is not claimed as derived, because that would smuggle the scheme anchor back in.)

4.4 Refute the wrong number rather than ship it

The decisive cultural move: the corpus refuted −2.818×10⁹⁴ GeV⁶ at decision grade rather than report it with soft caveats. A wrong magnitude — contaminated by R2 and resting on an ill-posed odd-D scheme object — is more dangerous to a program's credibility than an openly-stated gap. The honest edge is stronger than a fragile claim: "we found and refuted our own wrong number" is a trust-building result, exactly the opposite of the failure mode where a fabricated value (the protocol's "graviton weight 67" cautionary tale, or a back-solved magnitude) survives into a public artifact.

4.5 Locate the wall, then locate the cleanest handle on it (P0)

The 5C wall is global, but the program does not stop at "open." It isolates P0 — does gravity own an intrinsic minimal length or inherit the color R₀? — as the cleanest finite, falsifiable next-round handle, derived from the T-DEEP finding that R₀ is a pure color object (R₀/l_Planck ~ 194). This is the difference between an honest wall and a dead end: the wall is named, its shared-blocker identity (UQF-9) is recorded, and the single most tractable lever onto it is isolated. (We flag below that P0's status as "bounded" must be verified — a UQF-9 fixed point is itself a frontier wall — but the determination of P0 is a legitimate next-round target, not a unicorn.)

5. Evidence & reproducibility

5.1 The numbers and their exact sources

Every load-bearing number in this dossier, with the corpus file it traces to. (None is fabricated; uncomputed objects are marked OPEN.)

5.2 Certificates and their dispositions

• CERT_SCALEFREE_SECTOR_2026-06-25.md — DERIVED-GIVEN-E for the dimensionless/scale-free sector; explicitly excludes the dimensionful magnitude (labeled SCHEME-ANCHORED). Carries an honest open flag: a second Gilkey transcription (a6_sphere_crosscheck.py) reports FAIL on the Gilkey-formula side while agreeing on spectral values — an unresolved transcription contradiction that must be settled before any d=13 number is trusted.
• CERT_CURVATURE_DELTA_2026-06-25.md — VERIFIED/DISCHARGED for the 23/75 correction (four target-blind methods + from-scratch rebuild); the engine code fix is a proposed drop-in (REQUIRES-COUNTERSIGN), not applied; frozen .py files unmodified.
• GAP01_A6_RESULT_COMPLETE.md — the bulk-a₆ engine run; status capped at GAP01-A6 computed (consistency coefficient), "No promotion," MO-12 binding. The dimensionful number here is the one later refuted.

5.3 How a reader re-runs / re-derives

Reproduction is gated and target-blind by design. The corpus path is rendered/TOE/computational_runs_2026-06-23/a6_computation/:

1. Curvature (R2) check. Rebuild the K₆ = SU(3)/T² naturally-reductive curvature from the su(3) structure constants in both sign conventions; print first-Bianchi max-residual and |Riem|²/Scal². Expect: as-written 31/147 with Bianchi residual ≈ 0.143 (the bug); sign-flipped 23/75 with residual ≈ 3×10⁻¹⁶. (Scripts: verify_k6_curvature.py, k6_fd_nablaR.py; the certificate's session-scratch r3_verify.py.)
2. Bulk-delta check. Monkeypatch the frozen a6_compute.py::k6_riem_gilkey to the sign-flipped builder and re-run main(). Expect: corrected bulk ≈ −2.9957×10⁹⁴ GeV⁶, Δ = +6.305%, sign preserved, Berger gate True in both runs. Do not treat the resulting magnitude as derived (R3).
3. Scale-free check. Run a6_compute.py and a6_sphere_crosscheck.py; confirm 4/315, 74/63, 1139/63, 5/63 on the spectral side. Note the cross-script Gilkey-formula contradiction flagged above.

Anti-fabrication invariants (binding on any reproducer). Do not fabricate any entry of tr[a₆]; do not fabricate the sign C ~ −6.39; do not back-solve any scheme object to a desired magnitude; treat the frozen branch as READ-ONLY. A captured terminal log is not an independent reproduction (GAP01.md residual C: the bulk cross-check is a captured log, banked as owed).

6. Open gaps + closure path (the specialist work plan)

This is the most load-bearing section. For each open hole: (a) the precise open object; (b) why it's hard + the traps the verifier already caught; (c) exactly what closes it (target-blind, with success and refuting criteria); (d) machinery & inputs (exact paths); (e) leverage. Physics only; firewall observed. The holes are ordered by tractability, not by how much they move 5C — because the two holes that are 5C are global walls, and the honest tractable work all lives one floor below.

Triage note up front (verify before chasing). Two holes are flagged possible-unicorns: H1 (5C itself) and H3 (above-cutoff unitarity). A full UV completion of quantum gravity is not a bounded, pluggable task for anyone — marking it bounded:true is the BS-10 false-openness failure mode. The correct deliverable for H1/H3 is a WALL RECORD, not a closure attempt. The genuinely bounded work is H4/H5/H6 (one floor below).

H1 — 5C itself: a constructive UV completion / strong-coupling closure ⚠️ POSSIBLE-UNICORN (verify)

(a) Precise statement. A genuinely new construction on the frozen geometry: an in-category, target-blind non-perturbative quantization of the interacting graviton — e.g. a controlled asymptotic-safety fixed point beyond a declared truncation, or an explicit UV-complete graviton S-matrix above the cutoff. The geometry supplies the operator (L_grav + FP-ghost) but exhibits NO route across the wall (R8). This is the same object as UQF-9.

(b) Why it's hard / traps. This is a global open problem shared with all approaches; the geometry provides no constructive lever, and T-DEEP shows there is no finite-grain shortcut (R₀ is a pure color object, R₀/l_Planck ~ 194). Traps the verifier caught and you must not repeat: (i) marking this bounded:true with a what_would_close_it of "solve QG UV completion" is false-openness (BS-10) — it is a wall, not a horse; (ii) relabeling "a₆ computed" as "5C closed" (the signature mis-close — a₆ is necessary-not-sufficient, R7); (iii) asserting [ω] = 0 or naming an axiom for an un-run calculation (RELABEL_FAIL).

(c) What closes it. Three legitimate outcomes, not a clean binary: (1) a target-blind construction that exhibits a truncation-independent non-Gaussian fixed point (or a defensible non-continuum replacement) → constructive close; (2) a proof that no such fixed point exists on this geometry → refuting close (a negative is a valid terminus); (3) neither runnable here → emit a WALL RECORD: NAME the construction precisely, CLASSIFY it OPEN/wall (no known route), record the FAITHFUL-BRIDGE it hides, mark it SHARED with UQF-9 / UQF-14 / GRAVITON-5C in the shared-blocker ledger, and state the strongest honest endpoint. The WALL RECORD is the deliverable.

(d) Machinery & inputs. The FRG / asymptotic-safety toolkit (Wetterich equation; beyond-truncation stability); the frozen operator data (01_DOSSIER.md §1; 03_FROZEN_GEOMETRY_CONTEXT.md); the UQF-9 gate (GATE_REGRADE…2026-06-29.md UQF-9; the cost-floor/T-CONT material). Build the shared UQF-9 construction once, not this gate again.

(e) Leverage. Closing UQF-9 closes/lifts: this gate (5C), UQF-14 (above-cutoff unitarity), the 5B certificate-conditionality (it stops being conditional), and the Gap-01 sufficiency question. Maximum cross-gate leverage in the program — which is precisely why it is a shared frontier wall, not a per-gate plug.

H2 — P0: does gravity own an intrinsic shortest length, or inherit the color R₀?

(a) Precise statement. Determine, target-blind, whether the gravitational sector has its own minimal-length scale or inherits R₀ from the color sector. This is the newly isolated, cleanest UQF-9 sub-target — the most tractable handle on the 5C/UQF-9 wall.

(b) Why it's hard / traps. It is a sub-target of the UQF-9 frontier wall, so it is the weakest "bounded" item — treat its bounded status as provisional. The trap is to over-read a P0 determination as itself crossing the wall: it does not. A definite answer either opens a finite-grain route to 5C or confirms the wall — both are valid, and confirming the wall is not a failure.

(c) What closes it. A target-blind determination of the gravitational minimal-length disposition. Success criterion: a definite, reproducible answer (own-scale vs inherited-R₀) that does not back-solve to a desired disposition. Refuting-style outcome: the determination shows gravity inherits R₀ → confirms 5C has no finite-grain shortcut (consistent with T-DEEP), and the wall stands. This is a finite, falsifiable next-round target.

(d) Machinery & inputs. The T-DEEP result (R₀ a pure color/gauge object, R₀/l_Planck ~ 194); the cost-floor / granularity axiom and the Lorentz-scalar placement (T-LI: the floor sits on a scalar, picking no preferred frame — GATE_REGRADE…2026-06-29.md UQF-9 banked); the frozen geometry radii (CURVATURE_DATA.md §0, R₀).

(e) Leverage. P0 is the single highest-value bounded lever onto UQF-9/5C: a "gravity inherits R₀" answer sharpens the wall and confirms no-shortcut; a "gravity owns its own scale" answer would open a genuinely new finite-grain route. Either way it advances the whole UQF-9 cluster.

H3 — Graviton-sector unitarity above the cutoff (UQF-14 / R9) ⚠️ POSSIBLE-UNICORN (verify)

(a) Precise statement. Prove (or refute) unitarity of the graviton sector above the cutoff, where the graviton and KK tower go strongly coupled. Below the cutoff this is inherited standard QFT; above it, it is blocked by UQF-9.

(b) Why it's hard / traps. It is strictly downstream of H1 — it cannot be settled until the UV completion is, and it cannot be reframed as malformed without first proving P0. Trap: presenting above-cutoff unitarity as a ready falsifier independent of the UV completion (it is not).

(c) What closes it. Close UQF-9 (= close H1); above-cutoff unitarity then inherits its disposition (Froissart-bound / Landau-pole control rise with the UV-completion certificate). There is no independent close. Until then: WALL RECORD, downstream-of-H1.

(d) Machinery & inputs. Inherited from H1/UQF-9; the UQF-14 gate material (GATE_REGRADE…2026-06-29.md UQF-14: the banked perturbative + truncated-KK sub-row, held OPEN under the weakest-link min-rule).

(e) Leverage. None independent of H1; it rises automatically when H1 closes. Counted once with H1 in the shared-blocker ledger.

H4 — The leg below (5B): the d=13 graviton-minus-ghost a₆ vector + the positivity functional P (the real decision-grade work)

(a) Precise statement. Two coupled open objects: (i) the d=13 graviton-minus-ghost a₆ vector tr[a₆(L_grav)] projected on the cubic-curvature basis with K₆×S²×S¹_Y holonomy is uncomputed (R1; even the dimensionless coefficient sign C ~ −6.39 is unreproduced); (ii) the positivity functional P is itself unselected — three competing readings — so P(a₆) ≥ 0 is not yet a well-defined predicate (R4). The predicate is doubly open.

(b) Why it's hard / traps. The a₆ trace is "perturbative but heavy" — a definite but enormous symbolic computation over the ~46-term cubic Gilkey basis at d=13, reproducibility-gated. Traps the verifier explicitly caught: (i) do NOT fabricate the sign C ~ −6.39 — the EDGE bet hinges on it and it is currently unreproduced; the prior blind audit reproduced only the scalar sector (124/315), not the full graviton-ghost projection; (ii) do NOT axiomatize a pass of P(a₆) ≥ 0; (iii) do NOT present the falsifier as runnable while P is unselected — it is a falsifier-in-principle until P is fixed; (iv) the engine must be fixed for R2 first (the ~31% Riemann-norm deficit, now localized) or no magnitude is trustworthy.

(c) What closes it. Step 1: select the precise positivity functional P required for physical-Hilbert-space closure, plus its decision-grade adjudication rule (target-blind, value-free). Step 2: after fixing R2, compute the d=13 graviton-minus-ghost a₆ vector target-blind on the machine lane, reproducing the sign of C. Step 3: evaluate P(a₆). Success: P(a₆) ≥ 0 → lifts 5A/5B from CERTIFICATE-CONDITIONAL to certificate-grade conditional on UQF-9 (it never reaches 5C — R7). Refuting result: P(a₆) < 0 → REFUTES the linearized gate at decision grade (a valid, valuable close).

(d) Machinery & inputs. Gilkey (1995) Thm 4.8.16 / Vassilevich eq. (4.29) (GAP01_A6_RESULT_COMPLETE.md §1); the de-Donder operator + FP-ghost fiber trace ×65 (ibid. §1); the R2-corrected curvature builder (CERT_CURVATURE_DELTA_2026-06-25.md, the proposed sign-flipped Rop() drop-in); the engine a6_compute.py, cross-checks a6_sphere_crosscheck.py, derivative sector nablaR_K6.py / k6_fd_nablaR.py. Gap-04 reproducibility infrastructure is the stated prerequisite. Resolve the cross-script Gilkey-transcription contradiction (a6_compute.py PASS vs a6_sphere_crosscheck.py FAIL) before trusting any d=13 number.

(e) Leverage. This is the program's whole 5B credibility and the only decision-grade move available on the graviton cluster short of UQF-9. A pass strengthens 5A/5B; a violation is a clean refutation. It does not close 5C (firewall), but it is the highest-value bounded work in this dossier.

H5 — The scheme-anchored magnitude + the 124/315 independent reproduction (R3)

(a) Precise statement. The dimensionful a₆ magnitude is scheme/scale-anchored (rides one named, geometry-unfixed heat-kernel scheme object, the same family as c_loop / SG-7-δ), and is arguably ill-posed at odd D=13. Separately, the clean ratio 124/315 is DERIVED-PENDING — it must be reproduced by an engine that does not carry R2.

(b) Why it's hard / traps. The magnitude terminates on no measured invariant. Trap (κ³/π falsification test): a scheme object reverse-engineered to hit a desired magnitude is true-by-construction and relocates the residual, it does not close it. Trap (engine entanglement): 124/315 was produced by the same R2-carrying engine, so reproducing it through that engine proves nothing — it must be an independent route. Also carried honestly: 124/315 is metric-SELECTED (holds at the spectrum-demanded normalization), so it must not be advertised as the clean target-blind invariant.

(c) What closes it. (i) Reproduce 124/315 target-blind on a structurally independent engine (no routing through the R2 engine) → upgrades it from DERIVED-PENDING to a clean derived win. (ii) Either settle the shared one-loop heat-kernel scheme object target-blind (writable WITHOUT knowing the target magnitude) → if it lands, DERIVED-GIVEN-E; or name AND verify AXIOM-HEATKERNEL-SCHEME-OBJECT (value-free) → AXIOM-CLOSED-pending-verification (= still OPEN until verified). A reverse-engineered scheme RELOCATES — refuse it.

(d) Machinery & inputs. CERT_SCALEFREE_SECTOR_2026-06-25.md (the pole-structure / scheme analysis); the independent route hinted in the live brief (a6_07_k6_indep.py + a6_08_reconcile.py for the curvature ratio — a separate independent engine could be modeled on it for the color factor); the shared scheme-object family with SG-7-δ / c_loop.

(e) Leverage. Settling the scheme object is a shared close (it is the same object across c_loop / SG-7-δ / a₆) — high cross-gate value. Reproducing 124/315 independently is a clean, bounded win that also feeds H4 (a trustworthy color factor for the full vector).

H6 — The Z₂ orbifold-defect heat-kernel (R6, BLOCKED)

(a) Precise statement. The order-6 mixed Neumann+Dirichlet boundary heat-kernel coefficient for the S¹_Y/ℤ₂ orbifold fixed points (θ = 0, π) does not exist in the literature — the published boundary tower stops at a₅. Without it, no TOTAL a₆ (bulk + defect) can be emitted; the engine number is bulk-only.

(b) Why it's hard / traps. It is a genuine literature gap, not a fabrication-able piece. Trap: do NOT fabricate the defect total. The ℤ₂ defect is a separate summed term (Cheeger-cone / Gilkey orbifold-boundary object), never merged into a bulk Riemann monomial. (One honest note from the leg-below handoff: the defect's correct identity may be the Donnelly equivariant (1/2)c₃^γ rather than a missing boundary coefficient — a specialist must adjudicate this before declaring the object "missing-from-literature.")

(c) What closes it. A specialist construction of the higher-order S¹_Y/ℤ₂ boundary heat-kernel coefficient (likely a genuine new mathematical result), or a proof that the Donnelly equivariant form supplies it. Then assemble TOTAL = bulk + defect. If not obtainable: terminal as BLOCKED, exported to the tracked shared-blocker ledger (master-ledger B1). No total is to be fabricated.

(d) Machinery & inputs. CURVATURE_DATA.md §5.2 (the ℤ₂ defect named as a separate boundary term; routed to the Appendix-G heat-kernel boundary term); Gilkey 1995 (boundary/orbifold a_k); Cheeger (spectral geometry of cones); the shared-blocker build folder PER_GATE_DOSSIERS/SHARED_BLOCKER_BUILDS/B1_heatkernel/.

(e) Leverage. Shared with the Gap-01 a₆ cluster (the TOTAL a₆ is the same object across gates). Closing it makes a complete (bulk+defect) a₆ possible, which is a prerequisite for any total-magnitude statement — but note this remains 5B/Gap-01 work, never a 5C close.

7. Honest ceiling & scope

The ceiling, stated flat. UQF-5C is a serious candidate's honest acknowledgement of a global wall: the geometry supplies the operator, NOT a validated quantum theory of gravity. We do not claim 5C; we do not claim a UV completion; we do not claim a strong-coupling closure; we do not claim the interacting graviton S-matrix; we do not claim this geometry is the unique one that could supply a consistent spin-2 sector. The gate is OPEN (global wall), held, STATUS-UPGRADES:0.

Dissolved ≠ solved. The program's cost-floor / granularity axiom dissolves a class of continuum UV divergences (the a₈/a₁₀/… counterterm tower, the a → 0 idealization) on the path toward UQF-9 — but that is a scoped dissolution that explicitly leaves the finite a₆ obligation and the asymptotic-safety fixed point OPEN. Dissolving a class of artifacts is not solving the strong-coupling theory. We say so on the UQF-9 gate and we say so here.

Selection ≠ derivation; given-E ≠ derivation-of-E. The frozen background is READ-ONLY and selected, not proven uniquely forced. The linearized-GR claim (5A) holds given the observed gravity limit as input — it is not a from-nothing derivation of GR. The graviton operator is geometry-supplied, which is its load-bearing role and nothing more.

The unicorns, framed as shared ceilings (never as our weakness, never as proven).

• "No future theory could UV-complete gravity better than ours." A universal negative over all possible mathematics — unprovable for everyone. The honest ceiling: 5C is a global wall shared by all approaches, a limit on everyone's knowledge, not a hole in ours.
• "a₆ (one coefficient) settles the strong-coupling theory." The inverse over-claim — one heat-kernel coefficient cannot close an unbounded a₆ < a₈ < a₁₀ ladder. The scope firewall (R7) correctly forbids this; necessary-not-sufficient is the ceiling, not a hedge.
• "THE absolutely unique UV completion of the interacting graviton." Demanding proof of unique forcedness over the open-ended space of quantization schemes is a unicorn. The bounded claim — this geometry supplies the operator; the named missing object is the UV completion / UQF-9 — is the ceiling.

The anchors paid. The graviton magnitudes terminate on no measured invariant: the headline a₆ vector is uncomputed (R1), the dimensionful magnitude is scheme/scale-anchored (R3), the engine carried a ~31% curvature error (R2, corrected value 23/75 proved, engine fix owed). The one genuinely clean scale-free quantity is the dimensionless ratio 124/315 — and even that is DERIVED-PENDING independent target-blind reproduction. The structural graviton + ghost operator is geometry-supplied (5A, given-E). No measured-invariant witness exists for the a₆ magnitude, and any scheme reverse-engineered to a desired magnitude relocates rather than hardens (κ³/π falsification test).

The bright lines (never printed as proven). 5C is closed / the program has a UV completion — DENIED (OPEN, global wall). a₆ is computed therefore 5C is closed — DENIED (the signature mis-close; a₆ is the missing object for, never is, UV completion). The −2.818×10⁹⁴ GeV⁶ magnitude is a derived graviton result — DENIED (refuted at decision grade; scheme-anchored, R2-contaminated).

The strongest honest thing we can say. 5C is a global wall — and we have done the honest work around it that a serious program owes: named the wall precisely (a constructive non-perturbative UV completion = UQF-9), proved the curvature input (23/75), banked the scale-free ratio (124/315, pending independent reproduction), held the scope firewall, established via T-DEEP that there is no finite-grain shortcut (R₀/l_Planck ~ 194), isolated the single cleanest bounded handle (P0), and refuted our own wrong magnitude rather than ship it. The geometry supplies the operator. It does not certify the quantum theory. STATUS-UPGRADES:0; frozen branch dcc66f1b2685 / a5b1e6f9d951 READ-ONLY; given-E ≠ derivation of E; a captured log ≠ an independent reproduction; a₆ ≠ UV completion; nothing applied, nothing deployed.

Dossier built 2026-06-29 from the frozen corpus (graviton completion handoff, the a₆ computational runs, the 2026-06-25 certificates, and the 2026-06-29 gate-regrade brief). Our geometry (13D K₆ branch) only. Common material referenced to the corpus source-of-truth, not duplicated. Every number traceable to a named file in §5.